Let's go through the problem step by step. The image outlines the following tasks:

We are given a sample space $S = \{E_1, E_2, E_3, E_4, E_5, E_6, E_7\}$ with associated probabilities for each event:

$P(E_1) = 0.10, \quad P(E_2) = 0.20, \quad P(E_3) = 0.15, \quad P(E_4) = 0.25, \quad P(E_5) = 0.10, \quad P(E_6) = 0.05, \quad P(E_7) = 0.15$

You are also given three events:

• $A = \{E_1, E_4, E_5\}$
• $B = \{E_3, E_6, E_7\}$
• $C = \{E_4, E_5, E_7\}$

Task (a): Find $P(A)$, $P(B)$, and $P(C)$.

$P(A)$

$P(A) = P(E_1) + P(E_4) + P(E_5) = 0.10 + 0.25 + 0.10 = 0.45$

$P(B)$

$P(B) = P(E_3) + P(E_6) + P(E_7) = 0.15 + 0.05 + 0.15 = 0.35$

$P(C)$

$P(C) = P(E_4) + P(E_5) + P(E_7) = 0.25 + 0.10 + 0.15 = 0.50$

Task (b): What is $A \cup B$?

The union of $A$ and $B$ is the set of all elements that are in either $A$ or $B$:

$A \cup B = \{E_1, E_3, E_4, E_5, E_6, E_7\}$

Hence, $P(A \cup B)$ is:

$P(A \cup B) = P(E_1) + P(E_3) + P(E_4) + P(E_5) + P(E_6) + P(E_7) = 0.10 + 0.15 + 0.25 + 0.10 + 0.05 + 0.15 = 0.80$

Task (c): What is $A \cap B$?

The intersection of $A$ and $B$ is the set of elements that are in both $A$ and $B$:

$A \cap B = \varnothing$

Since the intersection is the empty set, $P(A \cap B) = 0$.

Task (d): Are events $A$ and $C$ mutually exclusive?

Two events are mutually exclusive if they have no elements in common, i.e., their intersection is the empty set. Let's find $A \cap C$:

$A \cap C = \{E_4, E_5\}$

Since $A \cap C \neq \varnothing$, the events $A$ and $C$ are not mutually exclusive.

If you would like further details or clarification on any of the steps, feel free to ask! Here are 5 follow-up questions related to this problem:

1. What is the probability of the complement of event $A$?
2. How do you calculate $P(A \cup C)$?
3. Are events $B$ and $C$ mutually exclusive?
4. Can we apply the addition rule for $P(A \cup B)$?
5. How would you approach finding $P(A \cap C)$ if more events were added to $A$ or $C$?

Tip: Remember, when calculating the union of two events, make sure not to double-count any elements that belong to both events!